Download The chemical history of molecules in circumstellar disks I: Ices

2
The chemical history of molecules in
circumstellar disks
I: Ices
R. Visser, E. F. van Dishoeck, S. D. Doty and C. P. Dullemond
Astronomy & Astrophysics, 2009, 495, 881
19
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
Abstract
Context. Many chemical changes occur during the collapse of a molecular cloud core to form a
low-mass star and the surrounding disk. One-dimensional models have been used so far to analyse
these chemical processes, but they cannot properly describe the incorporation of material into disks.
Aims. The goal of this chapter is to understand how material changes chemically as it is transported
from the cloud to the star and the disk. Of special interest is the chemical history of the material in
the disk at the end of the collapse.
Methods. We present a two-dimensional, semi-analytical model that, for the first time, follows the
chemical evolution from the pre-stellar core to the protostar and circumstellar disk. The model computes infall trajectories from any point in the cloud core and tracks the radial and vertical motion of
material in the viscously evolving disk. It includes a full time-dependent radiative transfer treatment
of the dust temperature, which controls much of the chemistry. We explore a small parameter grid to
understand the effects of the sound speed and the mass and rotation rate of the core. The freeze-out
and evaporation of carbon monoxide (CO) and water (H2 O), as well as the potential for forming
complex organic molecules in ices, are considered as important first steps towards illustrating the
full chemistry.
Results. Both species freeze out towards the centre before the collapse begins. Pure CO ice evaporates during the infall phase and readsorbs in those parts of the disk that cool below the CO desorption temperature of ∼18 K. H2 O remains solid almost everywhere during the infall and disk
formation phases and evaporates within ∼10 AU of the star. Mixed CO-H2 O ices are important
in keeping some solid CO above 18 K and in explaining the presence of CO in comets. Material that ends up in the planet- and comet-forming zones of the disk (∼5–30 AU from the star) is
predicted to spend enough time in a warm zone (several 104 yr at a dust temperature of 20–40 K)
during the collapse to form first-generation complex organic species on the grains. The dynamical
timescales in the hot inner envelope (hot core or hot corino) are too short for abundant formation of
second-generation molecules by high-temperature gas-phase chemistry.
20
2.1 Introduction
2.1 Introduction
The formation of low-mass stars and their planetary systems is a complex event, spanning
several orders of magnitude in temporal and spatial scales, and involving a wide variety
of physical and chemical processes. Thanks to observations, theory and computer simulations, the general picture of low-mass star formation is now understood (see reviews by
Shu et al. 1987, di Francesco et al. 2007, Klein et al. 2007, White et al. 2007 and Dullemond et al. 2007b). An instability in a cold molecular cloud core leads to gravitational
collapse. Rotation and magnetic fields cause a flattened density structure early on, which
evolves into a circumstellar disk at later times. The protostar continues to accrete matter
from the disk and the remnant envelope, while also expelling matter in a bipolar pattern.
Grain growth in the disk eventually leads to the formation of planets, and as the remaining dust and gas disappear, a mature solar system emerges. While there has been ample
discussion in the literature on the origin and evolution of grains in disks (see reviews by
Natta et al. 2007 and Dominik et al. 2007 or the discussion in Chapter 4), little attention
has so far been paid to the chemical history of the more volatile material in a two- or
three-dimensional setting.
Chemical models are required to understand the observations and develop the simulations (see reviews by Ceccarelli et al. 2007, Bergin et al. 2007 and Bergin & Tafalla 2007).
The chemistry in pre-stellar cores is relatively easy to model, because the dynamics and
the temperature structure are simpler before the protostar is formed than afterwards. A
key result from the pre-stellar core models is the depletion of many carbon-bearing species
towards the centre of the core (Bergin & Langer 1997, Lee et al. 2004).
The next step in understanding the chemical evolution during star formation is to
model the chemistry during the collapse phase (Ceccarelli et al. 1996, Rodgers & Charnley 2003, Doty et al. 2004, Lee et al. 2004, Garrod & Herbst 2006, Aikawa et al. 2008,
Garrod et al. 2008). All of the collapse chemistry models so far are one-dimensional, and
thus necessarily ignore the circumstellar disk. As the protostar turns on and heats up the
surrounding material, all models agree that frozen-out species return to the gas phase if
the dust temperature surpasses their evaporation temperature. The higher temperatures
can further drive a hot-core–like chemistry, and complex molecules may be formed if the
infall timescales are long enough.
If the model is expanded into a second dimension and the disk is included, the system
gains a large reservoir where infalling material from the cloud can be stored for a long time
(at least several 104 yr) before accreting onto the star. This can lead to further chemical
enrichment, especially in the warmer parts of the disk (Aikawa et al. 1997, 2008, Aikawa
& Herbst 1999, Willacy & Langer 2000, van Zadelhoff et al. 2003, Rodgers & Charnley
2003). The interior of the disk is shielded from direct irradiation by the star, so it is colder
than the disk’s surface and the remnant envelope. Hence, molecules that evaporated as
they fell in towards the star may freeze out again when they enter the disk. This was first
shown quantitatively by Brinch et al. (2008b, hereafter BWH08) using a two-dimensional
hydrodynamical simulation.
In addition to observations of nearby star-forming regions, the comets in our own solar
system provide a unique probe into the chemistry that takes place during star and planet
21
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
formation. The bulk composition of the cometary nuclei is believed to be mostly pristine,
closely reflecting the composition of the pre-solar nebula (Bockelée-Morvan et al. 2004).
However, large abundance variations have been observed between individual comets and
these remain poorly understood (Kobayashi et al. 2007). Two-dimensional chemical models may shed light on the cometary chemical diversity.
Two molecules of great astrophysical interest are carbon monoxide (CO) and water
(H2 O). They are the main reservoirs of carbon and oxygen and control much of the overall chemistry. CO is an important precursor for more complex molecules; for example,
solid CO can be hydrogenated to formaldehyde (H2 CO) and methanol (CH3 OH) at low
temperatures (Watanabe & Kouchi 2002, Fuchs et al. 2009). In turn, these two molecules
form the basis of even larger organic species like methyl formate (HCOOCH3 ; Garrod &
Herbst 2006, Garrod et al. 2008). The key role of H2 O in the formation of life on Earth
and potentially elsewhere is evident. If the entire formation process of low-mass stars
and their planets is to be understood, a thorough understanding of these two molecules is
essential.
This chapter is the first in a series of publications aiming to model the chemical evolution from the pre-stellar core to the disk phase in two dimensions, using a simplified,
semi-analytical approach for the dynamics of the collapsing envelope and the disk, but
including detailed radiative transfer for the temperature structure. The model follows individual parcels of material as they fall in from the cloud core into the disk. The gaseous
and solid abundances of CO and H2 O are calculated for each infalling parcel to obtain
global gas-ice profiles. The semi-analytical nature of the model allows for an easy exploration of physical parameters like the core’s mass and rotation rate, or the effective sound
speed. Tracing the temperature history of the infalling material provides a first clue into
the formation of more complex species. The model also offers some insight into the origin
of the chemical diversity in comets.
Section 2.2 contains a full description of the model. Results are presented in Sect.
2.3 and discussed in a broader astrophysical context in Sect. 2.4. Finally, conclusions are
drawn in Sect. 2.5.
2.2 Model
The physical part of our two-dimensional axisymmetric model describes the collapse of
an initially spherical, isothermal, slowly rotating cloud core to form a star and circumstellar disk. The collapse dynamics are taken from Shu (1977, hereafter S77), including the
effects of rotation as described by Cassen & Moosman (1981, hereafter CM81) and Terebey et al. (1984, hereafter TSC84). Infalling material hits the equatorial plane inside the
centrifugal radius to form a disk, whose further evolution is constrained by conservation
of angular momentum (Lynden-Bell & Pringle 1974). Some properties of the star and the
disk are adapted from Adams & Shu (1986) and Young & Evans (2005, hereafter YE05).
Magnetic fields are not included in our model. They are unlikely to affect the chemistry
directly and their main physical effect (causing a flattened density distribution; Galli &
Shu 1993) is already accounted for by the rotation of the core.
22
2.2 Model
Our model is an extension of the one used by Dullemond et al. (2006a) to study the
crystallinity of dust in circumstellar disks (see also Chapter 4). That model was purely
one-dimensional; our model treats the disk more realistically as a two-dimensional structure.
2.2.1 Envelope
The cloud core (or envelope) is taken to be a uniformly rotating singular isothermal sphere
at the onset of collapse. It has a solid-body rotation rate Ω0 and an r−2 density profile
(S77):
c2s
ρ0 (r) =
,
(2.1)
2πGr2
where G is the gravitational constant and cs the effective sound speed. Throughout this
thesis, r is used for the spherical radius and R for the cylindrical radius. Setting the outer
radius at renv , the total mass of the core is
M0 =
2c2s renv
.
G
(2.2)
After the collapse is triggered at the centre, an expansion wave (or collapse front)
travels outwards at the sound speed (S77, TSC84). Material inside the expansion wave
falls in towards the centre to form a protostar. The infalling material is deflected towards
the gravitational midplane by the core’s rotation. It first hits the midplane inside the
centrifugal radius (where gravity balances angular momentum; CM81), resulting in the
formation of a circumstellar disk (Sect. 2.2.2).
The dynamics of a collapsing singular isothermal sphere were computed by S77 in
terms of the non-dimensional variable x = r/cs t, with t the time after the onset of collapse. In this self-similar description, the head of the expansion wave is always at x = 1.
The density and radial velocity are given by the non-dimensional variables A and v, respectively. (S77 used α for the density, but our model already uses that symbol for the
viscosity in Sect. 2.2.2.) These variables are dimensionalised through
A(x)
,
4πGt2
(2.3)
ur (r, t) = cs v(x) .
(2.4)
ρ(r, t) =
Values for A and v are tabulated in S77.
CM81 and TSC84 analysed the effects of slow uniform rotation on the S77 collapse
solution, with the former focussing on the flow onto the protostar and the disk and the latter on what happens farther out in the envelope. In the axisymmetric TSC84 description,
the density and infall velocities depend on the time, the radius and the polar angle:
A(x, θ, τ)
,
4πGt2
(2.5)
ur (r, θ, t) = cs v(x, θ, τ) ,
(2.6)
ρ(r, θ, t) =
23
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
where τ = Ω0 t is the non-dimensional time. The polar velocity is given by
uθ (r, θ, t) = cs w(x, θ, τ) .
(2.7)
We solved the differential equations from TSC84 numerically to obtain solutions for A, v
and w.
The TSC84 solution breaks down around x = τ2 , so the CM81 solution is used inside
of this point. A streamline through a point (r, θ) effectively originated at an angle θ0 in
this description:
cos θ0 − cos θ Rc
−
= 0,
(2.8)
r
sin2 θ0 cos θ0
where Rc is the centrifugal radius,
Rc (t) =
1
cs m30 t3 Ω20 ,
16
(2.9)
with m0 a numerical factor equal to 0.975. The CM81 radial and polar velocity are
r
r
GM
cos θ
ur (r, θ, t) = −
1+
,
(2.10)
r
cos θ0
r
r
cos θ cos θ0 − cos θ
GM
1+
,
(2.11)
uθ (r, θ, t) =
r
cos θ0
sin θ
and the CM81 density is
ρ(r, θ, t) = −
−1
Ṁ Rc
P
(cos
θ
)
1
+
2
,
2
0
r
4πr2 ur
(2.12)
where P2 is the second-order Legendre polynomial and Ṁ = m0 c3s /G is the total accretion
rate from the envelope onto the star and disk (S77, TSC84). The primary accretion phase
ends when the outer shell of the envelope reaches the star and disk. This point in time
(tacc = M0 / Ṁ) is essentially the beginning of the T Tauri or Herbig Ae/Be phase, but it
does not yet correspond to a typical T Tauri or Herbig Ae/Be object (see Sect. 2.3.2).
The TSC84 and CM81 solutions do not reproduce the cavities created by the star’s
bipolar outflow, so they have to be put in separately. Outflows have been observed in
two shapes: conical and curved (Padgett et al. 1999). Both can be characterised by the
outflow opening angle, γ, which grows with the age of the object. Arce & Sargent (2006)
found a linear relationship in log-log space between the age of a sample of 17 young
stellar objects and their outflow opening angles. Some explanations exist for the outflow
widening in general, but it is not yet understood how γ(t) depends on parameters like the
initial cloud core mass and the sound speed. It is likely that the angle depends on the
relative age of the object rather than on the absolute age.
The purpose of our model is not to include a detailed description of the outflow cavity.
Instead, the outflow is primarily included because of its effect on the temperature profiles
(Whitney et al. 2003). Its opening angle is based on the fit by Arce & Sargent (2006)
24
2.2 Model
to their Fig. 5, but it is taken to depend on t/tacc rather than t alone. The outflow is also
kept smaller, which brings it closer to the Whitney et al. angles. Its shape is taken to be
conical. With the resulting formula,
log
γ(t)
t
,
= 1.5 + 0.26 log
deg
tacc
(2.13)
the opening angle is always 32◦ at t = tacc . The numbers in Eq. (2.13) are poorly constrained; however, the details of the outflow (both size and shape) do not affect the temperature profiles strongly, so this introduces no major errors in the chemistry results. The
outflow cones are filled with a constant mass of 0.002M0 at a uniform density, which
decreases to 103 –104 cm−3 at tacc depending on the model parameters. The outflow effectively removes about 1% of the initial envelope mass.
2.2.2 Disk
The rotation of the envelope causes the infalling material to be deflected towards the midplane, where it forms a circumstellar disk. The disk initially forms inside the centrifugal
radius (CM81), but conservation of angular momentum quickly causes the disk to spread
beyond this point. The evolution of the disk is governed by viscosity, for which our model
uses the common α prescription (Shakura & Sunyaev 1973). This gives the viscosity coefficient ν as
ν(R, t) = αcs,d H .
(2.14)
p
The sound speed in the disk, cs,d = kT m /µmp (with k the Boltzmann constant, mp the
proton mass and µ the mean molecular mass of 2.3 nuclei per hydrogen molecule), is
different from the sound speed in the envelope, cs , because the midplane temperature of
the disk, T m , varies as described in Hueso & Guillot (2005). The other variable from Eq.
(2.14) is the scale height:
cs,d
H(R, t) =
,
(2.15)
Ωk
where Ωk is the Keplerian rotation rate:
r
GM∗
,
(2.16)
Ωk (R, t) =
R3
with M∗ the stellar mass (Eq. (2.29)). The viscosity parameter α is kept constant at 10−2
(Dullemond et al. 2007b, Andrews & Williams 2007b).
Solving the problem of advection and diffusion yields the radial velocities inside the
disk (Dullemond et al. 2006a, Lynden-Bell & Pringle 1974):
3 ∂ √ uR (R, t) = − √
Σν R .
Σ R ∂R
(2.17)
∂Σ(R, t)
1 ∂
=−
(ΣRuR ) + S ,
∂t
R ∂R
(2.18)
The surface density Σ evolves as
25
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
where the source function S accounts for the infall of material from the envelope:
S (R, t) = 2Nρuz ,
(2.19)
with uz the vertical component of the envelope velocity field (Eqs. (2.6), (2.7), (2.10) and
(2.11)). The factor 2 accounts for the envelope accreting onto both sides of the disk and
the normalisation factor N ensures that the overall accretion rate onto the star and the disk
is always equal to Ṁ. Both ρ and uz in Eq. (2.19) are to be computed at the disk-envelope
boundary, which is defined at the end of this section.
As noted by Hueso & Guillot (2005), the infalling envelope material enters the disk
with a sub-Keplerian rotation rate, so, by conservation of angular momentum, it would
tend to move a bit farther inwards. Not taking this into account would artificially generate
angular momentum, causing the disk to take longer to accrete onto the star. As a consequence the disk has, at any given point in time, too high a mass and too large a radius.
Hueso & Guillot solved this problem by modifying Eq. (2.19) to place the material directly at the correct radius. However, this causes an undesirable discontinuity in the infall
trajectories. Instead, our model adds a small extra component to Eq. (2.17) for t < tacc :
r
3 ∂ √ GM
uR (R, t) = − √
Σν R − ηr
.
(2.20)
∂R
R
Σ R
The functional form of the extra term derives from the CM81 solution. A constant value
of 0.002 for ηr is found to reproduce very well the results of Hueso & Guillot. It also
provides a good match with the disk masses from Yorke & Bodenheimer (1999), YE05
and BWH08, whose models cover a wide range of initial conditions. The problem of
sub-Keplerian accretion is addressed in more detail in Chapter 4. There, we derive a
more rigorous solution and show that it does not affect the gas-ice results from the current
chapter in a significant way.
The disk’s inner radius is determined by the evaporation of dust as it gets heated above
a critical temperature by the stellar radiation (e.g., YE05):
s
L∗
,
(2.21)
Ri (t) =
4
4πσT evap
where σ is the Stefan-Boltzmann constant. The dust evaporation temperature, T evap, is
set to 2000 K. Taking an alternative value of 1500 or even 1200 K has no effect on our
results. The stellar luminosity, L∗ , is discussed in Sect. 2.2.3. Inward transport of material
at Ri leads to accretion from the disk onto the star:
Ṁd→∗ = −2πRi uR Σ ,
(2.22)
with the radial velocity, uR , and the surface density, Σ, taken at Ri . The disk gains mass
from the envelope at a rate Ṁe→d , so the disk mass evolves as
Z t
(2.23)
Md (t) =
Ṁe→d − Ṁd→∗ dt′ .
0
26
2.2 Model
We adopt a Gaussian profile for the vertical structure of the disk (Shakura & Sunyaev
1973):
!
z2
(2.24)
ρ(R, z, t) = ρc exp − 2 ,
2H
with z the height above the midplane. The scale height comes from Eq. (2.15) and the
midplane density is
Σ
ρc (R, t) = √ .
(2.25)
H 2π
Along with the radial motion (Eq. (2.20), taken to be independent of z), material also
moves vertically in the disk, as it must maintain the Gaussian profile at all times. To see
this, consider a parcel of material that enters the disk at time t at coordinates R and z into
a column with scale height H and surface density Σ. The column of material below the
parcel is
!
Z z
z
1
ρ(R, ζ, t)dζ = Σerf
(2.26)
√ ,
2
0
H 2
where erf is the error function. At a later time t′ , the entire column has moved to R′ and
has a scale height H ′ and a surface density Σ′ . The same amount of material must still be
below the parcel:
!
!
1
1 ′
z′
z
=
(2.27)
Σ erf
Σerf
√
√ .
2
2
H′ 2
H 2
Rearranging gives the new height of the parcel, z′ :
!#
"
√
z
′ ′ ′
′
−1 Σ
erf
z (R , t ) = H 2erf
,
√
Σ′
H 2
(2.28)
where erf −1 is the inverse of the error function. In the absence of vertical mixing, our
description leads to purely laminar flow.
The location of the disk-envelope boundary (needed, e.g., in Eq. (2.19)) is determined
in two steps. First, the surface is identified where the density due to the disk (Eq. (2.24))
equals that due to the envelope (Eqs. (2.5) and (2.12)). In order for accretion to take place
at a given point P1 on the surface, it must be intersected by an infall trajectory. Due to the
geometry of the surface, such a trajectory might also intersect the disk at a larger radius
P2 (Fig. 2.1). Material flowing in along that trajectory accretes at P2 instead of P1 . Hence,
the second step in determining the disk-envelope boundary consists of raising the surface
at “obstructed points” like P1 to an altitude where accretion can take place. The source
function is then computed at that altitude. Physically, this can be understood as follows:
the region directly above the obstructed points becomes less dense than what it would be
in the absence of a disk, because the disk also prevents material from reaching there. The
lower density above the disk reduces the downward pressure, so the disk puffs up and the
disk-envelope boundary moves to a higher altitude.
The infall trajectories in the vicinity of the disk are very shallow, so the bulk of the
material accretes at the outer edge. Because the disk quickly spreads beyond the centrifugal radius, much of the accretion occurs far from the star. In contrast, accretion in
27
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
Figure 2.1 – Schematic view of the disk-envelope boundary in the upper right quadrant of the (R, z)
plane. The black line indicates the surface where the density due to the disk equals that due to the
envelope. The grey line is the infall trajectory that would lead to point P1 . However, it already
intersects the disk at point P2 , so no accretion is possible at P1 . The disk-envelope boundary is
therefore raised at P1 until it can be reached freely by an infall trajectory.
one-dimensional collapse models occurs at or inside of Rc (see also Chapter 4). Our results are consistent with the hydrodynamical work of BWH08, where most of the infalling
material also hits the outer edge of a rather large disk. The large accretion radii lead to
weaker accretion shocks than commonly assumed (Sect. 2.2.5).
2.2.3 Star
The star gains material from the envelope and from the disk, so its mass evolves as
Z t
(2.29)
Ṁe→∗ + Ṁd→∗ dt′ .
M∗ (t) =
0
The protostar does not come into existence immediately at the onset of collapse; it is
preceded by the first hydrostatic core (FHC; Masunaga et al. 1998, Boss & Yorke 1995).
Our model follows YE05 and takes a lifetime of 2 × 104 yr and a size of 5 AU for the
FHC, independent of other parameters. After this stage, a rapid transition occurs from the
large FHC to a protostar of a few R⊙ :
s




t
−
20
000
yr
 + RPS
R∗ = (5 AU) 1 −
20 000 < t (yr) < 20 100 ,
(2.30)
∗
100 yr 
where RPS
∗ (ranging from 2 to 5 R⊙ ) is the protostellar radius from Palla & Stahler (1991).
For t > 2.01 × 104 yr, R∗ equals RPS
∗ . Our results are not sensitive to the exact values used
for the size and lifetime of the FHC or the duration of the FHC-protostar transition.
The star’s luminosity, L∗ , consists of two terms: the accretion luminosity, L∗,acc , dominant at early times, and the luminosity due to gravitational contraction and deuterium
burning, Lphot . The accretion luminosity comes from Adams & Shu (1986):
(
i
p
1 h
6u∗ − 2 + (2 − 5u∗ ) 1 − u∗ +
L∗,acc (t) = L0
6u∗
i)
p
h
η∗ 1 − (1 − ηd )Md 1 − (1 − ηd ) 1 − u∗ ,
(2.31)
2
28
2.2 Model
Figure 2.2 – Evolution of the mass of the envelope, star and disk (left panel) and the luminosity
(solid lines) and radius (dotted lines) of the star (right panel) for our standard model (black lines)
and our reference model (grey lines).
where L0 = GM Ṁ/R∗ (with M = M∗ + Md the total accreted mass), u∗ = R∗ /Rc , and
Z √
1 1/3 1 1 − u
Md = u ∗
du .
(2.32)
3
u4/3
u∗
Analytical solutions exist for the asymptotic cases of u∗ ≈ 0 and u∗ ≈ 1. For intermediate
values, the integral must be solved numerically. The efficiency parameters η∗ and ηd
in Eq. (2.31) have values of 0.5 and 0.75 for a 1 M⊙ envelope (YE05). The photospheric
luminosity is adopted from D’Antona & Mazzitelli (1994), using YE05’s method of fitting
and interpolating, including a time difference of 0.38 tacc (equal to the free-fall time)
between the onset of L∗,acc and L∗,phot (Myers et al. 1998). The sum of these two terms,
L∗ (t) = L∗,acc + L∗,phot , gives the total stellar luminosity.
Figure 2.2 shows the evolution of the stellar mass, luminosity and radius for our standard case of M0 = 1.0 M⊙ , cs = 0.26 km s−1 and Ω0 = 10−14 s−1 , and our reference
case of M0 = 1.0 M⊙ , cs = 0.26 km s−1 and Ω0 = 10−13 s−1 (Sect. 2.2.6). The transition
from the FHC to the protostar at t = 2 × 104 yr is clearly visible in the R∗ and L∗ profiles.
At t = tacc , there is no more accretion from the envelope onto the star, so the luminosity
decreases sharply.
The masses of the disk and the envelope are also shown in Fig. 2.2. Our disk mass of
0.43 M⊙ at t = tacc in the reference case is in excellent agreement with the value of 0.4
M⊙ found by BWH08 for the same parameters.
29
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
2.2.4 Temperature
The envelope starts out as an isothermal sphere at 10 K and it is heated up from the inside
after the onset of collapse. Using the star as the only photon source, we compute the dust
temperature in the disk and envelope with the axisymmetric three-dimensional radiative
transfer code RADMC (Dullemond & Dominik 2004a). Because of the high densities
throughout most of the system, the gas and dust are expected to be well coupled, and the
gas temperature is set equal to the dust temperature. Cosmic-ray heating of the gas is
included implicitly by setting a lower limit of 8 K in the dust radiative transfer results.
As mentioned in Sect. 2.2.1, the presence of the outflow cones has some effect on the
temperature profiles (Whitney et al. 2003). This is discussed further in Sect. 2.3.2.
2.2.5 Accretion shock
The infall of high-velocity envelope material into the low-velocity disk causes a J-type
shock. The temperature right behind the shock front can be much higher than what it
would be due to the stellar photons. Neufeld & Hollenbach (1994) calculated in detail the
relationship between the pre-shock velocities and densities (us and ns ) and the maximum
grain temperature reached after the shock (T d,s). A simple formula, valid for us < 70 km
s−1 , can be extracted from their Fig. 13:
T d,s ≈ (104 K)
p
0.21 agr
ns
us
−1
6
−3
0.1 µm
30 km s
10 cm
!−0.20
,
(2.33)
with agr the grain radius. The exponent p is 0.62 for us < 30 km s−1 and 1.0 otherwise.
The pre-shock velocities and densities are highest at early times, when accretion occurs close to the star and all ices would evaporate anyway. Important for our purposes
is the question whether the dust temperature due to the shock exceeds that due to stellar
heating. If all grains have a radius of 0.1 µm, as assumed in our model, this is not the case
for any of the material in the disk at tacc for either our standard or our reference model
(Fig. 2.3, cf. Simonelli et al. 1997).
In reality, the dust spans a range of sizes, extending down to a radius of about 0.005
µm. Small grains are heated more easily; 0.005-µm dust reaches a shock temperature
almost twice as high as does 0.1-µm dust (Eq. (2.33)). This is enough for the shock
temperature to exceed the radiative heating temperature in part of the sample in Fig. 2.3.
However, this has no effect on the CO and H2 O gas-ice ratios. In those parcels where
shock heating becomes important for small grains, the temperature from radiative heating
lies already above the CO evaporation temperature of about 18 K and the shock temperature remains below 60 K, which is not enough for H2 O to evaporate. Hence, shock heating
is not included in our model.
H2 O may also be removed from the grain surfaces in the accretion shock through
sputtering (Tielens et al. 1994, Jones et al. 1994). The material that makes up the disk at
the end of the collapse in our standard model experiences a shock of at most 8 km s−1 .
At that velocity, He+ , the most important ion for sputtering, carries an energy of 1.3 eV.
30
2.2 Model
Figure 2.3 – Dust temperature due to the accretion shock (vertical axis) and stellar radiation (horizontal axis) at the point of entry into the disk for 0.1-µm grains in a sample of several hundred
parcels in our standard (left) and reference (right) models. These parcels occupy positions from
R = 1 to 300 AU in the disk at tacc . Note the different scales between the two panels.
However, a minimum of 2.2 eV is required to remove H2 O (Bohdansky et al. 1980), so
sputtering is unimportant for our purposes.
Some of the material in our model is heated to more than 100 K during the collapse
(Fig. 2.12) or experiences a shock strong enough to induce sputtering. This material
normally ends up in the star before the end of the collapse, but mixing may keep some of
it in the disk. The possible consequences are discussed briefly in Sect. 2.4.4.
2.2.6 Model parameters
The standard set of parameters for our model corresponds to Case J from Yorke & Bodenheimer (1999), except that the solid-body rotation rate is reduced from 10−13 to 10−14
s−1 to produce a more realistic disk mass of 0.05 M⊙ , consistent with observations (e.g.,
Andrews & Williams 2007a,b). The envelope has an initial mass of 1.0 M⊙ and a radius
of 6700 AU, and the effective sound speed is 0.26 km s−1 .
The original Case J (with Ω0 = 10−13 s−1 ), which was also used in BWH08, is used
here as a reference model to enable a direct quantitative comparison of the results with an
independent method. This case results in a much higher disk mass of 0.43 M⊙ . Although
such high disk masses are not excluded by observations and theoretical arguments (Hartmann et al. 2006), they are considered less representative of typical young stellar objects
than the disks of lower mass.
The parameters M0 , cs and Ω0 are changed in one direction each to create a 23 parameter grid. The two values for Ω0 , 10−14 and 10−13 s−1 , cover the range of rotation rates
31
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
Table 2.1 – Summary of the parameter grid used in our model.a
Caseb
1
2
3 (std)
4
5
6
7 (ref)
8
Ω0
(s−1 )
10−14
10−14
10−14
10−14
10−13
10−13
10−13
10−13
cs
(km s−1 )
0.19
0.19
0.26
0.26
0.19
0.19
0.26
0.26
M0
(M⊙ )
1.0
0.5
1.0
0.5
1.0
0.5
1.0
0.5
tacc
(105 yr)
6.3
3.2
2.5
1.3
6.3
3.2
2.5
1.3
τads
(105 yr)
14.4
3.6
2.3
0.6
14.4
3.6
2.3
0.6
Md
(M⊙ )
0.22
0.08
0.05
0.001
0.59
0.25
0.43
0.16
a
Ω0 : solid-body rotation rate; cs : effective sound speed; M0 : initial core mass; tacc : accretion time; τads :
adsorption timescale for H2 O at the edge of the initial core; Md : disk mass at tacc .
b
Case 3 is our standard parameter set and Case 7 is our reference set.
observed by Goodman et al. (1993). The other variations are chosen for their opposite
effect: a lower sound speed gives a more massive disk, and a lower initial core mass gives
a less massive disk. The full model is run for each set of parameters to analyse how the
chemistry can vary between different objects. The parameter grid is summarised in Table
2.1. Our standard set is Case 3 and our reference set is Case 7.
Table 2.1 also lists the accretion time and the adsorption timescale for H2 O at the edge
of the initial core. For comparison, Evans et al. (2009) found a median timescale for the
embedded phase (Class 0 and I) of 5.4 × 105 yr from observations. It should be noted
that the end point of our model (tacc ) is not yet representative of a typical T Tauri disk
(see Sect. 2.3.2). Nevertheless, it allows an exploration of how the chemistry responds to
plausible changes in the environment.
2.2.7 Adsorption and desorption
The adsorption and desorption of CO and H2 O are solved in a Lagrangian frame. When
the time-dependent density, velocity and temperature profiles have been calculated, the
envelope is populated by a number of parcels of material (typically 12 000) at t = 0.
They fall in towards the star or disk according to the velocity profiles. The density and
temperature along each parcel’s infall trajectory are used as input to solve the adsorptiondesorption balance. Both species start fully in the gas phase. The envelope is kept static
for 3 × 105 yr before the onset of collapse to simulate the pre-stellar core phase. This is
the same value as used by Rodgers & Charnley (2003) and BWH08, and it is consistent
with recent observations by Enoch et al. (2008). The amount of gaseous material left over
near the end of the pre-collapse phase is also consistent with observations, which show
that the onset of H2 O ice formation is around an AV of 3 mag (Whittet et al. 2001). In
six of our eight parameter sets, the adsorption timescales of H2 O at the edge of the cloud
core are shorter than the combined collapse and pre-collapse time (Table 2.1), so all H2 O
32
2.3 Results
is expected to freeze out before entering the disk. Because of the larger core size, the
adsorption timescales are much longer in Cases 1 and 5, and some H2 O may still be in
the gas phase when it reaches the disk.
No chemical reactions are included other than adsorption and thermal desorption, so
the total abundance of CO and H2 O in each parcel remains constant. The adsorption rate
in cm−3 s−1 is taken from Charnley et al. (2001):
s
Tg
−18
3 −1/2 −1
Rads (X) = (4.55 × 10 cm K
s )nH ng (X)
,
(2.34)
M(X)
where nH is the total hydrogen density, T g the gas temperature, ng (X) the gas-phase abundance of species X and M(X) its molecular weight. The numerical factor assumes unit
sticking efficiency, a grain radius of 0.1 µm and a grain abundance xgr of 10−12 with respect to H2 .
The thermal desorption of CO and H2 O is a zeroth-order process:
#
"
Eb (X)
,
(2.35)
Rdes (X) = (1.26 × 10−21 cm2 )nH f (X)ν(X) exp −
kT d
where T d is the dust temperature and
"
#
ns (X)
f (X) = min 1,
,
Nb ngr
(2.36)
with ns (X) the solid abundance of species X and Nb = 106 the typical number of binding
sites per grain. The numerical factor in Eq. (2.35) assumes the same grain properties as in
Eq. (2.34). The pre-exponential factor, ν(X), and the binding energy, Eb (X)/k, are set to
7 × 1026 cm−2 s−1 and 855 K for CO and to 1 × 1030 cm−2 s−1 and 5773 K for H2 O (Fraser
et al. 2001, Bisschop et al. 2006).
Using a single Eb (CO) value means that all CO evaporates at the same temperature.
This would be appropriate for a pure CO ice, but not for a mixed CO-H2 O ice as is
likely to form in reality. During the warm-up phase, part of the CO is trapped inside the
H2 O ice until the temperature becomes high enough for the H2 O to evaporate. Recent
laboratory experiments suggest that CO desorbs from a CO-H2 O ice in several discrete
steps (Collings et al. 2004; Fayolle et al. in prep.). We simulate this in some model runs
with four “flavours” of CO ice, each with a different Eb (CO) value (Viti et al. 2004). For
each flavour, the desorption is assumed to be zeroth order. The four-flavour model is
summarised in Table 2.2.
2.3 Results
Results are presented in this section for our standard and reference models (Cases 3 and
7) as described in Sect. 2.2.6. These cases are compared to the other parameter sets in
Sect. 2.4.1. The appendix at the end of this chapter describes a formula to estimate the
33
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
Table 2.2 – Binding energies and desorbing fractions for the four-flavour CO evaporation model.a
Flavour
1
2
3
4
Eb (CO)/k (K)b
855
960
3260
5773
Fractionc
0.350
0.455
0.130
0.065
a
Based on Viti et al. (2004).
The rates for Flavours 1–3 are computed from Eq. (2.35) with X = CO. The rate for Flavour 4 is equal to
the H2 O desorption rate.
c
Fractions of adsorbing CO: 35% of all adsorbing CO becomes Flavour 1, and so on.
b
disk formation efficiency, defined as Md /M0 at the end of the collapse phase, based on a
fit to our model results. The results from this section are partially revisited in Chapters 3
and 4, where we use a different solution to the problem of sub-Keplerian accretion.
2.3.1 Density profiles and infall trajectories
In our standard model (Case 3), the envelope collapses in 2.5 ×105 yr to give a star of 0.94
M⊙ and a disk of 0.05 M⊙ . The remaining 0.01 M⊙ has disappeared through the bipolar
outflow. The centrifugal radius in our standard model at tacc is 4.9 AU, but the disk has
spread to 400 AU at that time due to angular momentum redistribution. The densities in
the disk are high: more than 109 cm−3 at the midplane inside of 120 AU (Fig. 2.4, top)
and more than 1014 cm−3 near 0.3 AU. The corresponding surface densities of the disk
are 2.0 g cm−2 at 120 AU and 660 g cm−2 at 0.3 AU.
Due to the higher rotation rate, our reference model (Case 7) gets a much higher disk
mass: 0.43 M⊙ . This value is consistent with the mass of 0.4 M⊙ reported by BWH08.
Overall, the reference densities from our semi-analytical model (Fig. 2.4, bottom) compare well with those from their more realistic hydrodynamical simulations; the differences
are generally less than a factor of two.
In both cases, the disk first emerges at 2 × 104 yr, when the FHC contracts to become
the protostar, but it is not until a few 104 yr later that the disk becomes visible on the scale
of Fig. 2.4. The regions of high density (nH > 105 –106 cm−3 ) are still contracting at that
time, but the growing disks eventually cause them to expand again.
Material falls in along nearly radial streamlines far out in the envelope and deflects
towards the midplane closer in. When a parcel enters the disk, it follows the radial motion
Figure 2.4 – Total density at four time steps for our standard model (Case 3; top) and our reference
model (Case 7; bottom). The notation a(b) denotes a×10b . The density contours increase by factors
of ten going inwards; the 105 -cm−3 contours are labelled in the standard panels and the 106 -cm−3
contours in the reference panels. The white curves indicate the surface of the disk as defined in
Sect. 2.2.2 (only visible in three panels). Note the different scale between the two sets of panels.
34
2.3 Results
35
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
caused by the viscous evolution and accretion of more material from the envelope. At
any time, conservation of angular momentum causes part of the disk to move inwards
and part of it to move outwards. An individual parcel entering the disk may move out
for some time before going farther in. This takes the parcel through several density and
temperature regimes, which may affect the gas-ice ratios or the chemistry in general. The
back-and-forth motion occurs especially at early times, when the entire system changes
more rapidly than at later times. The parcel motions are visualised in Figs. 2.5 and 2.6,
where infall trajectories are drawn for 24 parcels ending up at one of eight positions at
tacc : at the midplane or near the surface at radial distances of 10, 30, 100 and 300 AU.
Only parcels entering the disk before t ≈ 2 × 105 yr in our standard model or t ≈ 1 × 105
yr in our reference model undergo the back-and-forth motion. The parcels ending up near
the midplane all enter the disk earlier than the ones ending up at the surface.
Accretion from the envelope onto the disk occurs in an inside-out fashion. Because
of the geometry of the disk (Fig. 2.1), a lot of the material enters near the outer edge and
prevents the older material from moving farther out. Our flow inside the disk is purely
laminar, so some material near the midplane does move outwards underneath the newer
material at higher altitudes.
Because of the low rotation rate in our standard model, the disk does not really begin
to build up until 1.5 × 105 yr (0.6 tacc ) after the onset of collapse. In addition, most of the
early material to reach the disk makes it to the star before the end of the accretion phase,
so the disk at tacc consists only of material from the edge of the original cloud core (Fig.
2.7, top).
The disk in our reference model, however, begins to form right after the FHC-protostar
transition at 2 × 104 yr. As in the standard model, a layered structure is visible in the disk,
but it is more pronounced here. At the end of the collapse, the midplane consists mostly
of material that was originally close to the centre of the envelope (Fig. 2.7, bottom). The
surface and outer parts of the disk are made up primarily of material from the outer parts
of the envelope. This was also reported by BWH08.
2.3.2 Temperature profiles
When the star turns on at 2 × 104 yr, the envelope quickly heats up and reaches more than
100 K inside of 10 AU. As the disk grows, its interior is shielded from direct irradiation
and the midplane cools down again. At the same time, the remnant envelope material
above the disk becomes less dense and warmer. As in Whitney et al. (2003), the outflow
has some effect on the temperature profile. Photons emitted into the outflow can scatter
and illuminate the disk from the top, causing a higher disk temperature beyond R ≈ 200
AU than if there were no outflow cone. At smaller radii, the disk temperature is lower
than in a no-outflow model. Without the outflow, the radiation would be trapped in the
inner envelope and inner disk, increasing the temperature at small radii.
At t = tacc in our standard model, the 100- and 18-K isotherms (where H2 O and pure
CO evaporate) intersect the midplane at 20 and 2000 AU (Fig. 2.8, top). The disk in our
reference model is denser and therefore colder: it reaches 100 and 18 K at 5 and 580 AU
on the midplane (Fig. 2.8, bottom). Our radiative transfer method is a more rigorous way
36
2.3 Results
to obtain the dust temperature than the diffusion approximation used by BWH08, so our
temperature profiles are more realistic than theirs.
Compared to typical T Tauri disk models (e.g., D’Alessio et al. 1998, 1999, 2001),
our standard disk at tacc is warmer. It is 81 K at 30 AU on the midplane, while the closest
model from the D’Alessio catalogue is 28 K at that point. If our model is allowed to
run beyond tacc , part of the disk accretes further onto the star. At t = 4 tacc (1 Myr), the
disk mass goes down to 0.03 M⊙ . The luminosity of the star decreases during this period
(D’Antona & Mazzitelli 1994), so the disk cools down: the midplane temperature at 30
AU is now 42 K. Meanwhile, the dust is likely to grow to larger sizes, which would
further decrease the temperatures (D’Alessio et al. 2001). Hence, it is important to realise
that the normal end point of our models does not represent a “mature” T Tauri star and
disk as typically discussed in the literature.
2.3.3 Gas and ice abundances
Our two species, CO and H2 O, begin entirely in the gas phase. They freeze out during
the static pre-stellar core phase from the centre outwards due to the density dependence
of Eq. (2.34). After the pre-collapse phase of 3 × 105 yr, only a few tenths of per cent of
each species is still in the gas phase at 3000 AU. About 30% remains in the gas phase at
the edge of the envelope.
Up to the point where the disk becomes important and the system loses its spherical
symmetry, our model gives the same results as the one-dimensional collapse models: the
temperature quickly rises to a few tens of K in the collapsing region, driving most of the
CO into the gas phase, but keeping H2 O on the grains.
As the disk grows in mass, it provides an increasingly large body of material that
is shielded from the star’s radiation, and that is thus much colder than the surrounding
envelope. However, the disk in our standard model never gets below 18 K before the end
of the collapse (Sect. 2.3.2), so CO remains in the gas phase (Fig. 2.9, top). Note that
trapping of CO in the H2 O ice is not taken into account here; this possibility is discussed
in Sect. 2.4.3.
The disk in our reference model is more massive and therefore colder. After about
5 × 104 yr, the outer part drops below 18 K. CO arriving in this region readsorbs onto the
grains (Fig. 2.10, top). Another 2 × 105 yr later, at t = tacc , 19% of all CO in the disk is
in solid form. Moving out from the star, the first CO ice is found at the midplane at 400
AU. The solid fraction gradually increases to unity at 600 AU. At R = 1000 AU, nearly
all CO is solid up to an altitude of 170 AU. The solid and gaseous CO regions meet close
to the 18-K surface. The densities throughout most of the disk are high enough that once
a parcel of material goes below the CO desorption temperature, all CO rapidly disappears
from the gas. The exception to this rule occurs at the outer edge, near 1500 AU, where
the adsorption and desorption timescales are longer than the dynamical timescales of the
infalling material. Small differences between the trajectories of individual parcels then
cause some irregularities in the gas-ice profile.
The region containing gaseous H2 O is small at all times during the collapse. At t =
tacc , the snow line (the transition of H2 O from gas to ice) lies at 15 AU at the midplane
37
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
Figure 2.5 – Infall trajectories for parcels in our standard model (Case 3) ending up at the midplane
(left) or near the surface (right) at radial positions of 10, 30, 100 and 300 AU (dotted lines) at
t = tacc . Each panel contains trajectories for three parcels, which are illustrative for material ending
up at the given location. Trajectories are only drawn up to t = tacc . Diamonds indicate where each
parcel enters the disk; the time of entry is given in units of 105 yr. Note the different scales between
some panels.
38
2.3 Results
Figure 2.6 – Same as Fig. 2.5, but for our reference model (Case 7).
39
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
Figure 2.7 – Position of parcels of material in our standard model (Case 3; top) and our reference
model (Case 7; bottom) at the onset of collapse (t = 0) and at the end of the collapse phase (t =
tacc ). The parcels are coloured according to their initial position. The layered accretion is most
pronounced in our reference model. The grey parcels from t = 0 end up in the star or disappear in
the outflow. Note the different spatial scale between the two panels of each set; the small boxes in
the left panels indicate the scales of the right panels.
Figure 2.8 – Dust temperature, as in Fig. 2.4. Contours are drawn at 100, 60, 50, 40, 35, 30, 25, 20,
18, 16, 14 and 12 K. The 40- and 20-K contours are labelled in the standard and reference panels,
respectively. The 18- and 100-K contours are drawn in grey. The white curves indicate the surface
of the disk as defined in Sect. 2.2.2 (only visible in four panels).
40
2.3 Results
41
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
Figure 2.9 – Gaseous CO as a fraction of the total CO abundance (top) and idem for H2 O (bottom)
at two time steps for our standard model (Case 3). The black curves indicate the surface of the disk
(only visible in two panels). The black area near the pole is the outflow, where no abundances are
computed. Note the different spatial scale between the two panels of each set; the small box in the
left CO panel indicates the scale of the H2 O panels.
in our standard model (Fig. 2.9, bottom). The surface of the disk holds gaseous H2 O out
to R = 41 AU, and overall 13% of all H2 O in the disk is in the gas phase. This number
is much lower in the colder disk of our reference model: only 0.4%. The snow line now
lies at 7 AU and gaseous H2 O can be found out to 17 AU in the disk’s surface layers (Fig.
2.10, bottom).
Using the adsorption-desorption history of all the individual infalling parcels, the original envelope can be divided into several chemical zones. This is trivial for our standard
model. All CO in the disk is in the gas phase and it has the same qualitative history: it
freezes out before the onset of collapse and quickly evaporates as it falls in. H2 O also
freezes out initially and only returns to the gas phase if it reaches the inner disk.
42
2.3 Results
Figure 2.10 – Same as Fig. 2.9, but for our reference model (Case 7). The CO gas fraction is plotted
on a larger scale and at two additional time steps.
Our reference model has the same general H2 O adsorption-desorption history, but it
shows more variation for CO, as illustrated in Fig. 2.11. For the red parcels in that figure,
more than half of the CO always remains on the grains after the initial freeze-out phase.
On the other hand, more than half of the CO comes off the grains during the collapse for
43
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
Figure 2.11 – Same as Fig. 2.7, but only for our reference model (Case 7) and with a different colour
scheme to denote the CO adsorption-desorption behaviour. In all parcels, CO adsorbs during the
pre-collapse phase. Red parcels: CO remains adsorbed; green parcels: CO desorbs and readsorbs;
pink parcels: CO desorbs and remains desorbed; blue parcels: CO desorbs, readsorbs and desorbs
once more. The fraction of gaseous CO in each type of parcel as a function of time is indicated
schematically in the inset in the right panel. The grey parcels from t = 0 end up in the star or
disappear in the outflow. In our standard model (Case 3), all CO in the disk at the end of the
collapse phase is in the gas phase and it all has the same qualitative adsorption-desorption history,
equivalent to the pink parcels.
the green parcels, but it freezes out again inside the disk. The pink parcels, ending up in
the inner disk or in the upper layers, remain warm enough to keep CO off the grains once
it first evaporates. The blue parcels follow a more erratic temperature profile, with CO
evaporating, readsorbing and evaporating a second time. This is related to the back-andforth motion of some material in the disk (Fig. 2.6).
2.3.4 Temperature histories
The proximity of the CO and H2 O gas-ice boundaries to the 18- and 100-K surfaces
indicates that the temperature is primarily responsible for the adsorption and desorption.
At nH = 106 cm−3 , adsorption and desorption of CO are equally fast at T d = 18 K
(a timescale of 9 × 103 yr). For a density a thousand times higher or lower, the dust
temperature only has to increase or decrease by 2–3 K to maintain kads = kdes .
The exponential temperature dependence in the desorption rate (Eq. (2.35)) also holds
for other species than CO and H2 O, as well as for the rates of some chemical reactions.
Hence, it is useful to compute the temperature history for infalling parcels that occupy a
certain position at tacc . Figures 2.12 and 2.13 show these histories for the parcels from
Figs. 2.5 and 2.6. These parcels end up at the midplane or near the surface of the disk
at radial distances of 10, 30, 100 and 300 AU. Parcels ending up inside of 10 AU have
44
2.4 Discussion
a temperature history very similar to those ending up at 10 AU, except that the final
temperature of the former is higher.
Each panel in Figs. 2.12 and 2.13 contains the history of three parcels ending up
close to the desired position. The qualitative features are the same for all parcels. The
temperature is low while a parcel remains far out in the envelope. As it falls in with
an ever higher velocity, there is a temperature spike as it traverses the inner envelope,
followed by a quick drop once it enters the disk. Inward radial motion then leads to a
second temperature rise; because of the proximity to the star, this one is higher than the
first. For most parcels in Figs. 2.12 and 2.13, the second peak does not occur until long
after tacc . In all cases, the shock encountered upon entering the disk is weak enough that
it does not heat the dust to above the temperature caused by the stellar photons (Fig. 2.3).
Based on the temperature histories, the gas-ice transition at the midplane would lie
inside of 10 AU for H2 O and beyond 300 AU for CO in both our models. This is indeed
where they were found to be in Sect. 2.3.3. The transition for a species with an intermediate binding energy, such as H2 CO, is then expected to be between 10 and 100 AU, if its
abundance can be assumed constant throughout the collapse.
The dynamical timescales of the infalling material before it enters the disk are between 104 and 105 yr. The timescales decrease as it approaches the disk, due to the
rapidly increasing velocities. Once inside the disk, the material slows down again and
the dynamical timescales return to 104 –105 yr. The adsorption timescales for CO and
H2 O are initially a few 105 yr, so they exceed the dynamical timescale before entering the
disk. Depletion occurs nonetheless because of the pre-collapse phase of 3 × 105 yr. The
higher densities in the disk cause the adsorption timescales to drop to 100 yr or less. If
the temperature approaches (or crosses) the desorption temperature for CO or H2 O, the
corresponding desorption timescale becomes even shorter than the adsorption timescale.
Overall, the timescales for these specific chemical processes (adsorption and desorption)
in the disk are shorter by a factor of 1000 or more than the dynamical timescales.
At some final positions, there is a wide spread in the time that the parcels spend at a
given temperature. This is especially true for parcels ending up near the midplane inside
of 100 AU in our reference model. All of the midplane parcels ending up near 10 AU
exceed 18 K during the collapse; the first one does so at 3.5 × 104 yr after the onset of
collapse, the last one at 1.6 × 105 yr. Hence, some parcels at this final position spend
more than twice as long above 18 K than others. This does not appear to be relevant for
the gas-ice ratio, but it is important for the formation of more complex species (Garrod &
Herbst 2006, Garrod et al. 2008). This is discussed in more detail in Sect. 2.4.2.
2.4 Discussion
2.4.1 Model parameters
When the initial conditions of our model are modified (Sect. 2.2.6), the qualitative chemistry results do not change. In Cases 3, 4 and 8, the entire disk at tacc is warmer than
18 K, and it contains no solid CO. In the other cases, the disk provides a reservoir of
45
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
Figure 2.12 – Temperature history for parcels in our standard model (Case 3) ending up at the
midplane (left) or near the surface (right) at radial positions of 10, 30, 100 and 300 AU at t = tacc .
The colours correspond to Fig. 2.5. The dotted lines are drawn at T d = 18 K and t = tacc . Note the
different vertical scales between some panels.
46
2.4 Discussion
Figure 2.13 – Same as Fig. 2.12, but for our reference model (Case 7). The colours correspond to
Fig. 2.6.
47
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
relatively cold material where CO, which evaporates early on in the collapse, can return
to the grains. H2 O can only desorb in the inner few AU of the disk and remnant envelope.
Figures 2.14 and 2.15 show the density and dust temperature at tacc for each parameter
set; our standard and reference models are the two panels on the second row (Case 3 and
7). Several trends are visible:
• With a lower sound speed (Cases 1, 2, 5 and 6), the overall accretion rate ( Ṁ) is
smaller so the accretion time increases (tacc ∝ c−3
s ). The disk can now grow larger
and more massive. In our standard model, the disk is 0.05 M⊙ at tacc and extends to
about 400 AU radially. Decreasing the sound speed to 0.19 km s−1 (Case 1) results
in a disk of 0.22 M⊙ and nearly 2000 AU. The lower accretion rate also reduces the
stellar luminosity. These effects combine to make the disk colder in the low-cs cases.
• With a lower rotation rate (Cases 1–4), the infall occurs in a more spherically symmetric fashion. Less material is captured in the disk, which remains smaller and less
massive. From our reference to our standard model, the disk mass goes from 0.43 to
0.05 M⊙ and the radius from 1400 to 400 AU. The stronger accretion onto the star
causes a higher luminosity. Altogether, this makes for a small, relatively warm disk
in the low-Ω0 cases.
• With a lower initial mass (Cases 2, 4, 6 and 8), there is less material to end up on the
disk. The density profile is independent of the mass in a Shu-type collapse (Eq. (2.1)),
so the initial mass is lowered by taking a smaller envelope radius. The material from
the outer parts of the envelope is the last to accrete and is therefore more likely to end
up in the disk. If the initial mass is halved relative to our standard model (as in Case
4), the resulting disk is only 0.001 M⊙ and 1 AU. Our reference disk goes from 0.43
M⊙ and 1400 AU to 0.16 M⊙ and 600 AU (Cases 7 and 8). The luminosity at tacc is
lower in the high-M0 cases and the cold part of the disk (T d < 18 K) has a somewhat
larger relative size.
Dullemond et al. (2006a) noted that accretion occurs closer to the star for a slowly
rotating core than for a fast rotating core, resulting in a larger fraction of crystalline dust
in the former case. The same effect is seen here, but overall the accretion takes place
farther from the star than in Dullemond et al. (2006a). This is due to our taking into
account the vertical structure of the disk. Our gaseous fractions in the low-Ω0 disks are
higher than in the high-Ω0 disks (consistent with a higher crystalline fraction), but not
because material enters the disk closer to the star. Rather, as mentioned above, the larger
gas content comes from the higher temperatures throughout the disk. A more detailed
discussion of the crystallinity in disks can be found in Chapter 4.
Combining the density and the temperature, the fractions of cold (T d < 18 K), warm
(T d > 18 K) and hot (T d > 100 K) material in the disk can be computed. The warm and
hot fractions are listed in Table 2.3 along with the fractions of gaseous CO and H2 O in
the disk at tacc . Across the parameter grid, 23–100% of the CO is in the gas, along with
0.3–100% of the H2 O. This includes Case 4, which only has a disk of 0.0014 M⊙ . If that
one is omitted, at most 13% of the H2 O in the disk at tacc is in the gas. The gaseous H2 O
fractions for Cases 1, 2, 6, 7 and 8 (at most a few per cent) are quite uncertain, because the
48
2.4 Discussion
Table 2.3 – Summary of properties at t = tacc for our parameter grid.a
Case
1
2
3 (std)
4
5
6
7 (ref)
8
a
Md /M b
0.22
0.15
0.05
0.003
0.59
0.50
0.43
0.33
fwarm c
0.69
0.94
1.00
1.00
0.15
0.34
0.83
1.00
fhot c
0.004
0.035
0.17
1.00
0.0001
0.0004
0.003
0.028
fgas (CO)d
0.62
0.93
1.00
1.00
0.23
0.27
0.81
1.00
fgas (H2 O)d
0.028
0.020
0.13
1.00
0.11
0.02
0.004
0.003
These results are for the one-flavour CO desorption model.
b
The fraction of the disk mass with respect to the total accreted mass (M = M∗ + Md ).
c
d
The fractions of warm (T d > 18 K) and hot (T d > 100 K) material with respect to the entire disk. The
warm fraction also includes material above 100 K.
The fractions of gaseous CO and H2 O with respect to the total amounts of CO and H2 O in the disk.
model does not have sufficient resolution in the inner disk to resolve these small amounts.
They may be lower by up to a factor of ten or higher by up to a factor of three.
There is good agreement between the fractions of warm material and gaseous CO. In
Case 5, about a third of the CO gas at tacc is gas left over from the initial conditions, due to
the long adsorption timescale for the outer part of the cloud core. This is also the case for
the majority of the gaseous H2 O in Cases 1, 5 and 6. For the other parameter sets, fhot and
fgas (H2 O) are the same within the error margins. Overall, the results from the parameter
grid show once again that the adsorption-desorption balance is primarily determined by
the temperature, and that the adsorption-desorption timescales are usually shorter than the
dynamical timescales.
By comparing the fraction of gaseous material at the end of the collapse to the fraction
of material above the desorption temperature, the history of the material is disregarded.
For example, some of the cold material was heated above 18 K during the collapse, and
CO desorbed before readsorbing inside the disk. This may affect the CO abundance if the
model is expanded to include a full chemical network. In that case, the results from Table
2.3 only remain valid if the CO abundance is mostly constant throughout the collapse.
The same caveat holds for H2 O. This point is explored in more detail in Chapter 3.
2.4.2 Complex organic molecules
A full chemical network, including grain-surface reactions, is required to analyse the gas
and ice abundances of more complex species. While this will be a topic for a future publication, the current CO and H2 O results, combined with recent other work, can already
provide some insight.
Complex organic species can be divided into two categories: first-generation species
that are formed on the grain surfaces during the initial warm-up to ∼40 K, and second49
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
Figure 2.14 – Total density at t = tacc for our parameter grid (Table 2.1). The rotation rates (log Ω0
in s−1 ), sound speeds (km s−1 ) and initial masses (M⊙ ) are indicated. The contours increase by
factors of ten going inwards; the 106 -cm−3 contour is labelled in each panel. The white curves
indicate the surfaces of the disks; the disk for Case 4 is too small to be visible.
50
2.4 Discussion
Figure 2.15 – Dust temperature, as in Fig. 2.14. The temperature contours are drawn at 100, 60,
40, 30, 25, 20, 18, 16, 14 and 12 K from the centre outwards. The 20-K contour is labelled in each
panel and the 18-K contours are drawn as grey lines.
51
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
generation species that are formed in the warm gas phase when the first-generation species
have evaporated (Herbst & van Dishoeck 2009). Additionally, CH3 OH may be considered
a zeroth-generation complex organic because it is already formed efficiently during the
pre-collapse phase (Garrod & Herbst 2006). Its gas-ice ratio should be similar to that of
H2 O, due to the similar binding energies.
Larger first-generation species such as methyl formate (HCOOCH3 ) can be formed
on the grains if material spends at least several 104 yr at 20–40 K. The radicals involved
in the surface formation of HCOOCH3 (HCO and CH3 O) are not mobile enough at lower
temperatures and are not formed efficiently enough at higher temperatures. A low surface abundance of CO (at temperatures above 18 K) does not hinder the formation of
HCOOCH3 : HCO and CH3 O are formed from reactions of OH and H with H2 CO, which
is already formed at an earlier stage and which does not evaporate until ∼40 K (Garrod &
Herbst 2006). Cosmic-ray–induced photons are available to form OH from H2 O even in
the densest parts of the disk and envelope (Shen et al. 2004).
As shown in Sect. 2.3.4, many of the parcels ending up near the midplane inside of
∼300 AU in our standard model spend sufficient time in the required temperature regime
to allow for efficient formation of HCOOCH3 and other complex organics. Once formed,
these species are likely to assume the same gas-ice ratios as H2 O and the smaller organics. They evaporate in the inner 10–20 AU, so in the absence of mixing, complex organics
would only be observable in the gas phase close to the star. The Atacama Large Millimeter/submillimeter Array (ALMA), currently under construction, will be able to test this
hypothesis.
The gas-phase route towards complex organics involves the hot inner envelope (T d >
100 K), also called the hot core or hot corino in the case of low-mass protostars (Ceccarelli
2004, Bottinelli et al. 2004, 2007). Most of the ice evaporates here and a rich chemistry
can take place if material spends at least several 103 yr in the hot core (Charnley et al.
1992). However, the material in the hot inner envelope in our model is essentially in
freefall towards the star or the inner disk, and its transit time of a few 100 yr is too short
for complex organics to be formed abundantly (see also Schöier et al. 2002). Additionally,
the total mass in this region is very low: about a per cent of the disk mass. In order to
explain the observations of second-generation complex molecules, there has to be some
mechanism to keep the material in the hot core for a longer time. Alternatively, it has
recently been suggested that molecules typically associated with hot cores may in fact
form on the grain surfaces as well (Garrod et al. 2008).
2.4.3 Mixed CO-H2O ices
In the results presented in Sect. 2.3, all CO was taken to desorb at a single temperature.
In a more realistic approach, some of it would be trapped in the H2 O ice and desorb at
higher temperatures. This was simulated with four “flavours” of CO ice, as summarised in
Table 2.2. With our four-flavour model, the global gas-ice profiles are mostly unchanged.
All CO is frozen out in the sub-18 K regions and it fully desorbs when the temperature
goes above 100 K. Some 10 to 20% remains in the solid phase in areas of intermediate
temperature. In our standard model, the four-flavour variety has 15% of all CO in the disk
52
2.4 Discussion
at tacc on the grains, compared to 0% in the one-flavour variety. In our reference model,
the solid fraction increases from 19 to 33%.
The grain-surface formation of H2 CO, CH3 OH, HCOOCH3 and other organics should
not be very sensitive to these variations. H2 CO and CH3 OH are already formed abundantly before the onset of collapse, when the one- and four-flavour models predict equal
amounts of solid CO. H2 CO is then available to form HCOOCH3 (via the intermediates
HCO and CH3 O) during the collapse. The higher abundance of solid CO at 20–40 K in
the four-flavour model could slow down the formation of HCOOCH3 somewhat, because
CO destroys the OH needed to form HCO (Garrod & Herbst 2006). H2 CO evaporates
around 40 K, so HCOOCH3 cannot be formed efficiently anymore above that temperature. On the other hand, if a multiple-flavour approach is also employed for H2 CO, some
of it remains solid above 40 K, and HCOOCH3 can continue to be produced. Overall,
then, the multiple-flavour desorption model is not expected to cause large variations in
the abundances of these organic species compared to the one-flavour model.
2.4.4 Implications for comets
Comets in our solar system are known to be abundant in CO and they are believed to
have formed between 5 and 30 AU in the circumsolar disk (Bockelée-Morvan et al. 2004,
Kobayashi et al. 2007). However, the dust temperature in this region at the end of the
collapse is much higher than 18 K for all of our parameter sets. This raises the question
of how solid CO can be present in the comet-forming zone.
One possible answer lies in the fact that even at t = tacc , our objects are still very
young. As noted in Sect. 2.3.2, the disks cool down as they continue to evolve towards
“mature” T Tauri systems. Given the right set of initial conditions, this may bring the
temperature below 18 K inside of 30 AU. However, there are many T Tauri disk models in
the literature where the temperature at those radii remains well above the CO evaporation
temperature (e.g., D’Alessio et al. 1998, 2001). Specifically, models of the minimummass solar nebula (MMSN) predict a dust temperature of ∼40 K at 30 AU (Lecar et al.
2006).
A more plausible solution is to turn to mixed ices. At the temperatures computed for
the comet-forming zone of the MMSN, 10–20% of all CO may be trapped in the H2 O
ice. Assuming typical CO-H2 O abundance ratios, this is entirely consistent with observed
cometary abundances (Bockelée-Morvan et al. 2004).
Large abundance variations are possible for more complex species, due to the different
densities and temperatures at various points in the comet-forming zone in our model, as
well as the different density and temperature histories for material ending up at those
points. This seems to be at least part of the explanation for the chemical diversity observed
in comets. Our current model is extended in Chapter 3 to include a full gas-phase chemical
network to analyse these variations and compare them against cometary abundances.
The desorption and readsorption of H2 O in the disk-envelope boundary shock has
been suggested as a method to trap noble gases in the ice and include them in comets
(Owen et al. 1992, Owen & Bar-Nun 1993). As shown in Sects. 2.2.5 and 2.3.4, a number
of parcels in our standard model are heated to more than 100 K just prior to entering
53
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
the disk. However, these parcels end up in the disk’s surface. Material that ends up at
the midplane, in the comet-forming zone, never gets heated above 50 K. Vertical mixing,
which is ignored in our model, may be able to bring the noble-gas–containing grains down
into the comet-forming zone.
Another option is episodic accretion, resulting in temporary heating of the disk (Sect.
2.4.5). In the subsequent cooling phase, noble gases may be trapped as the ices reform.
The alternative of trapping the noble gases already in the pre-collapse phase is unlikely.
This requires all the H2 O to start in the gas phase and then freeze out rapidly. However,
in reality (contrary to what is assumed in our model) it is probably formed on the grain
surfaces by hydrogenation of atomic oxygen, which would not allow for trapping of noble
gases.
2.4.5 Limitations of the model
The physical part of our model is known to be incomplete and this may affect the chemical
results. For example, our model does not include radial and vertical mixing. Semenov
et al. (2006) and Aikawa (2007) recently showed that mixing can enhance the gas-phase
CO abundance in the sub-18 K regions of the disk. Similarly, there could be more H2 O gas
if mixing is included. This would increase the fractions of CO and H2 O gas listed in Table
2.3. The gas-phase abundances can also be enhanced by allowing for photodesorption of
the ices in addition to the thermal desorption considered here (Shen et al. 2004, Öberg
et al. 2007, 2009b,c). Mixing and photodesorption can each increase the total amount of
gaseous material by up to a factor of two. The higher gas-phase fractions are mostly found
in the regions where the temperature is a few degrees below the desorption temperature
of CO or H2 O.
Accretion from the envelope onto the star and disk occurs in our model at a constant
rate Ṁ until all of the envelope mass is gone. However, the lack of widespread red-shifted
absorption seen in interferometric observations suggests that the infall may stop already
at an earlier time (Jørgensen et al. 2007). This would reduce the disk mass at tacc . The
size of the disk is determined by the viscous evolution, which would probably not change
much. Hence, if accretion stops or slows down before tacc , the disk would be less dense
and therefore warmer. It would also reduce the fraction of disk material where CO never
desorbed, because most of that material comes from the outer edge of the original cloud
core (Fig. 2.11). Both effects would increase the gas-ice ratios of CO and H2 O.
Our results are also modified by the likely occurrence of episodic accretion (Kenyon
& Hartmann 1995, Evans et al. 2009). In this scenario, material accretes from the disk
onto the star in short bursts, separated by intervals where the disk-to-star accretion rate
is a few orders of magnitude lower. The accretion bursts cause luminosity flares, briefly
heating up the disk before returning to an equilibrium temperature that is lower than in
our models. This may produce a disk with a fairly large ice content for most of the
time, which evaporates and readsorbs after each accretion episode. The consequences for
complex organics and the inclusion of various species in comets are unclear.
54
2.5 Conclusions
2.5 Conclusions
This chapter presents the first results from a two-dimensional, semi-analytical model that
simulates the collapse of a molecular cloud core to form a low-mass protostar and its surrounding disk. The model follows individual parcels of material from the core into the star
or disk and also tracks their motion inside the disk. It computes the density and temperature at each point along these trajectories. The density and temperature profiles are used
as input for a chemical code to calculate the gas and ice abundances for carbon monoxide
(CO) and water (H2 O) in each parcel, which are then transformed into global gas-ice profiles. Material ending up at different points in the disk spends a different amount of time
at certain temperatures. These temperature histories provide a first look at the chemistry
of more complex species. The main results from this chapter are as follows:
• Both CO and H2 O freeze out towards the centre of the core before the onset of collapse. As soon as the protostar turns on, a fraction of the CO rapidly evaporates, while
H2 O remains on the grains. CO returns to the solid phase when it cools below 18 K
inside the disk. Depending on the initial conditions, this may be in a small or a large
fraction of the disk (Sect. 2.3.3).
• All parcels that end up in the disk have the same qualitative temperature history (Fig.
2.12). There is one temperature peak just before entering the disk, when material
traverses the inner envelope, and a second one (higher than the first) when inward
radial motion brings the parcel closer to the star. In some cases, this results in multiple
desorption and adsorption events during the parcel’s infall history (Sect. 2.3.4).
• Material that originates near the midplane of the initial core remains at lower temperatures than does material originating from closer to the poles. As a result, the chemical
content of the material from near the midplane is less strongly modified during the
collapse than the content of material from other regions (Fig. 2.11). The outer part of
the disk contains the chemically most pristine material, where at most only a small
fraction of the CO ever desorbed (Sect. 2.3.3).
• A higher sound speed results in a smaller and warmer disk, with larger fractions of
gaseous CO and H2 O at the end of the envelope accretion. A lower rotation rate has
the same effect. A higher initial mass results in a larger and colder disk, and smaller
gaseous CO and H2 O fractions (Sect. 2.4.1).
• The infalling material generally spends enough time in a warm zone (20–40 K) for
first-generation complex organic species to be formed abundantly on the grains (Fig.
2.12). Large differences can occur in the density and temperature histories for material
ending up at various points in the disk. These differences allow for spatial abundance
variations in the complex organics across the entire disk. This appears to be at least
part of the explanation for the cometary chemical diversity (Sects. 2.4.2 and 2.4.4).
• Complex second-generation species are not formed abundantly in the warm inner
envelope (the hot core or hot corino) in our model, due to the combined effects of the
dynamical timescales and low mass fraction in that region (Sect. 2.4.2).
• The temperature in the disk’s comet-forming zone (5–30 AU from the star) lies well
above the CO desorption temperature, even if effects of grain growth and continued
55
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
disk evolution are taken into account. Observed cometary CO abundances can be
explained by mixed ices: at temperatures of several tens of K, as predicted for the
comet-forming zone, CO can be trapped in the H2 O ice at a relative abundance of a
few per cent (Sect. 2.4.4).
Appendix: Disk formation efficiency
The results from our parameter grid can be used to derive the disk formation efficiency,
ηdf , as a function of the sound speed, cs , the solid-body rotation rate, Ω0 , and the initial
core mass, M0 . This efficiency can be defined as the fraction of M0 that is in the disk at
the end of the collapse phase (t = tacc ) or as the mass ratio between the disk and the star
at that time. The former is used in this appendix.
In order to cover a wider range of initial conditions, the physical part of our model
was run on a 93 grid. The sound speed was varied from 0.15 to 0.35 km s−1 , the rotation
rate from 10−14.5 to 10−12.5 s−1 and the initial core mass from 0.1 to 2.1 M⊙ . The resulting
ηdf at t = tacc were fitted to
"
#
Md
log(Ω0 /s−1 )
ηdf =
(2.37)
= g1 + g2
M0
−13
with
g 1 = k1 + k2
"
log(Ω0 /s−1 )
−13
g 2 = k5 + k6
"
#q1
+ k3
"
#q
q2
M0 3
cs
+
k
,
4
M⊙
0.2 km s−1
"
#
#
M0
cs
log(Ω0 /s−1 )
+
k
+ k7
.
8
−13
M⊙
0.2 km s−1
(2.38)
(2.39)
Equation (2.37) can give values lower than 0 or larger than 1. In those cases, it should be
interpreted as being 0 or 1.
The best-fit values for the coefficients ki and the exponents qi are listed in Table 2.4.
The absolute and relative difference between the best fit and the model data have a root
mean square (rms) of 0.04 and 5%. The largest absolute and relative difference are 0.20
and 27%. The fit is worst for a high core mass, a low sound speed and an intermediate
rotation rate, as well as for a low core mass, an intermediate to high sound speed and a
high rotation rate.
Figure 2.16 shows the disk formation efficiency as a function of the rotation rate,
including the fit from Eq. (2.37). The efficiency is roughly a quadratic function in log Ω0 ,
but due to the narrow dynamic range of this variable, the fit appears as straight lines.
Furthermore, the efficiency is roughly a linear function in cs and a square root function in
M0 .
56
2.5 Conclusions
Table 2.4 – Coefficients and exponents for the best fit for the disk formation efficiency.
Coefficient
k1
k2
k3
k4
Value
2.08
0.020
0.035
0.914
Coefficient
k5
k6
k7
k8
Value
−0.106
−1.539
−0.470
−0.344
Exponent
q1
q2
q3
Value
0.236
0.255
0.537
Figure 2.16 – Disk formation efficiency as a function of the solid-body rotation rate. The model
values are plotted as symbols and the fit from Eq. (2.37) as lines. The different values of the sound
speed are indicated by colours and the different values of the initial core mass are indicated by
symbols and line types, with the solid lines corresponding to the asterisks, the dotted lines to the
diamonds and the dashed lines to the triangles.
57